29 June 2016

Introduction

  • design efficient data collection methods
    • minimize costs of time/money
    • maximize information

- A primary sample of \(n_1\) units is selected from \(N\) units, based on some probability sampling scheme

Introduction

  • Variables: occupancy, abundance
  • Set up 1346 1-m\(^2\) plots at 7 habitat patches from May-July 2012 in Northeastern Florida

Adaptive Cluster Sampling (ACS)

Proposed in Thompson, S. (1990). Adaptive cluster sampling. Journal of the American Statistical Association, 85(412), 1050–1059.

Design:

  • A primary sample of \(n_1\) units is selected from \(N\) units, based on some probability sampling scheme

- A primary sample of \(n_1\) units is selected from \(N\) units, based on some probability sampling scheme - A primary sample of \(n_1\) units is selected from \(N\) units, based on some probability sampling scheme probability sampling scheme

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Adaptive Cluster Sampling (ACS)

Design:

  • Adaptive: A secondary sample is selected if primary units satisfy a condition (e.g., cacti are present)
  1. Adjacent units sampled in the four cardinal directions

- A primary sample of \(n_1\) units is selected from \(N\) units, based on some probability sampling scheme - A primary sample of \(n_1\) units is selected from \(N\) units, based on some probability sampling scheme - A primary sample of \(n_1\) units

Adaptive Cluster Sampling (ACS)

Design:

  1. For neighboring units whose values satisfy the condition, their neighbors are also surveyed

  2. Cluster Sampling: Sampling of neighboring units continues until no additional units satisfy the condition

- A primary sample of \(n_1\) units is selected from \(N\) units, based on some probability sampling scheme - A primary sample of \(n_1\) units is selected from \(N\) units, based on some probability sampling scheme

Adaptive Cluster Sampling (ACS)

Advantages

  • Best for studies involving populations of rare and clustered individuals
  • Compared to designs such as simple random sampling:
    • Increased sampling efficiency
    • Reduced variance

Disadvantage

  • Final sample size is unknown at the start of sampling

Restricted ACS (RACS)

  • same procedure as ACS, except that we restrict the number of neighboring units to \(q\) (in our example, \(q = 12\))

Estimators

  • The unbiased (for ACS) Horvitz-Thompson estimator of the population mean of variable \(x\) is

\[\bar{x}_{HT} = \frac{1}{N}\sum_{i=1}^{v} (x_i J_i)/\pi_i\]

  • \(N\) = population size, \(n_1\) = primary sample size
  • \(v\) = the number of unique units in the sample
  • \(J_i\) is an indicator variable, equalling 1 if the \(i\)^th unit in the sample satisfies the condition or was included in the primary sample and 0 otherwise
  • inclusion probability: \(\pi_i = 1 - \binom{N - m_i}{n_1} \Big/ \binom{N}{n_1}\), where \(m_i\) is the number of units in the network \(\psi_i\)

Simulations

  • Objective: Explore properties of the RACS design by bootstrapping populations created with a dataset of two cactus species and their associated insects.
  • randomly sampled the survey data to generate 6 populations

Hypotheses

Horvitz-Thompson estimators will be

  1. biased for the variable on which the RACS criterion is based
    • i.e., cactus presence
  2. unbiased for variables that
    • are not used to generate the RACS criterion
    • are uncorrelated with the variable on which the RACS criterion is based
    • i.e., insect presence, evidence of insect damage

Simulations

  • We used cactus presence as our criterion to initiate cluster sampling

  • We ran 5000 simulations of the RACS and ACS designs per

    • population
    • primary sample size (20, 30, 40, 75, and 100)
  • For each simulation we estimated:

    • O. stricta, O. pusilla, and cactus occupancy
    • insect occupancy given cactus presence

Simulations

  • relative bias of the mean:

\(B_{\bar{x}_{HT,i}} = (\bar{x}_{HT,i} - \mu)/\mu\)

  • \(\mu\) is the true population mean
  • \(\bar{x}_{HT,i}\) Horvitz-Thompson mean of variable \(x\) from the \(i\)^th simulation

  • mean percent relative bias for the \(S\) simulations

\(\text{mean}(B_{\bar{x}_{HT}}) = 100 \times S^{-1} \sum_{i=1}^{S} B_{\bar{x}_{HT,i}}\)

Results - Final Sample Sizes

Results - Cactus Occupancy

  • with RACS, we underestimate occupancy (negative percent relative bias)
  • this underestimate increases with cactus occupancy

Results - Cactus Occupancy

  • Variability decreases with increased primary sample size and occupancy

Moth Occupancy on O. stricta

  • bias decreases with increasing primary sample size
  • noise suggests that something other than just occupancy (e.g., spatial structure) is influencing the bias

Conclusions (So Far)

  • Compared to ACS, RACS generates smaller final sample sizes, particularly for populations with relatively high occupancy
  • confirmed that estimate of cactus occupancy is biased
    • may need to fix calculation so that bias does not increase with \(n_1\)
  • with larger sample sizes, our estimates of moth occupancy are less biased

Bias Correction Techniques

  • correct bias by correcting inclusion/intersection probabilities
  • ways to do this:
    • analytical - come up with a formula based on average behavior
    • use nonparametric technique such as bootstrapping
      • keep track of how many times a unit is sampled we can estimate the probabilities

Additional Work

  • we have also looked at bias for:
    • variance
    • insect damage occupancy
    • max. plant height
    • percentage of plot covered by cacti
  • we have created a R package, "ACSampling," that we will release as a public Github repository

Future Work

  • Estimate bias of the variance
  • Run more simulations
  • Test bias reduction techniques
  • compare efficiency of RACS, ACS, and SRSWOR
    • primary sample size = 40 + RACS = 40 vs. SRSWOR = 80?

Acknowledgements

  • R.D. Holt

Data Collection

  • Guana Tolomato Matanzas National Estuarine Research Reserve
  • Katie Petrinec
  • Matt Welsh
  • Polly Harding, undergrad. volunteer

Funding

  • QSE3 IGERT (NSF Award # 0801544)
  • Cactus and Succulent Society of America
  • Garden Club of America/Center for Plant Conservation
  • University of Florida Graduate Student Council
  • Department of Biology
  • Botanical Society of America
  • Florida Native Plant Society